Darboux transformation,infinite conservation laws,and exact solutions for the nonlocal Hirota equation with variable coefficients

This article presents the construction of a nonlocal Hirota equation with variable coefficients and its Darboux transformation.Using zero-seed solutions,1-soliton and 2-soliton solutions of the equation are constructed through the Darboux transformation,along with the expression for N-soliton solutions.Influence of coefficients that are taken as a function of time instead of a constant,i.e.,coefficient functionδ(t),on the solutions is investigated by choosing the coefficient functionδ(t),and the dynamics of the solutions are analyzed.This article utilizes the Lax pair to construct infinite conservation laws and extends it to nonlocal equations.The study of infinite conservation laws for nonlocal equations holds significant implications for the integrability of nonlocal equations.

Energy and first law of thermodynamics for Born-Infeld-anti-de-Sitter black hole

We calculate the local energy and the energy density of the Reisner-Norstrom-anti-de-Sitter black hole, study the first law of thermodynamics and show the Smarr formula for the Born-Infeld-anti-de-Sitter black hole. Applying the first law of thermodynamics to the black hole region, we analyse the three energy exchange processes between the black hole region and the outer and the inner regions.

Hierarchy of Lax Integrable Lattice Equations, Darboux Transformation and Conservation Laws

A discrete isospectral problem and the associated hierarchy of Lax integrable lattice equations were investigated. A Darboux transformation for the discrete spectral problem was found. Finally, an infinite number of conservation laws were given for the corresponding hierarchy.

NEW CONSERVATION LAWS OF ENERGY AND C-D INEQUALITIES IN CONTINUA WITHOUT MICROSTRUCTURE

Fundamental laws and balance equations as well as C-D inequalities in continuum mechanics are carefully restudied, incompleteness of existing balance laws of angular momentum and conservation laws of energy as well as C-D inequalities are pointed out, and finally new and more general conservation laws of energy and corresponding balance equations of energy as well as C-D inequalities in local and nonlocal asymmetric continua are presented.

NEW CONSERVATION LAWS OF ENERGY AND C-D INEQUALITIES IN CONTINUA WITH MICROSTRUCTURE

Existing fundamental laws, balance equations and Clausius-Duhem inequalities in continua with microstructure are systematically restudied, and the incomplete formulations of conservation laws of energy and related C-D inequalities are pointed out. Some remarks on existing results are made, and new conservation laws of energy and related C-D inequalities are presented.

Infinitely Many Conservation Laws of a Differential-difference Equation and Backlund Transformation

In this paper, with the help of the Lax representation, we show the existence of infinitely many conservation laws for a differential-difference equation,which is one of the Ladic-Ablowitz hierarchy, and the conservation density and the associated flux are given formularlly. We also demonstrate the relation between a continuous partial differential equation and the differential-difference equation, and give Backlund transformation for the former.

Connection between the Principles of Thermodynamics and the Conservation Laws: Physical Meaning of the Principles of Thermodynamics

It has been shown that the first principle of thermodynamics follows from the conservation laws for energy and linear momentum. And the second principle of thermodynamics follows from the first principle of thermodynamics under realization of the integrating factor (namely, temperature) and is a conservation law. The significance of the first principle of thermodynamics consists in the fact that it specifies the thermodynamic system state, which depends on interaction between conservation laws and is non-equilibrium due to a non-commutativity of conservation laws. The realization of the second principle of thermodynamics points to a transition of the thermodynamic system state into a locally-equilibrium state. Phase transitions are examples of such transitions.

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅷ)—PRINCIPLE OF TOTAL WORK AND ENERGY

Theoretical incompleteness of the existing conservation laws of energy for polar continuum mechanics is further clarified. For completeness, the principles of total work and energy and of total work and energy of incremental rate type are postulated. Via total variations of the former and the latter of them, the principles of virtual displacement and microrotation & stress and couple stress as well as virtual velocity and angular velocity & stress rate and couple stress rate are immediately obtained, respectively. From these principles all balance equations and boundary conditions for micropolar mechanics are naturally and simultaneously deduced. The essential differences between the nontraditional results obtained in this paper and the existing conservation laws of energy are expounded.

BACKLUND TRANSFORMATION, NONLINEAR SUPERPOSITION FORMULAE AND INFINITE CONSERVED LAWS OF BENJAMIN EQUATION

Backlund transformation, exact solitary wave solutions, nonlinear supperposi tion formulae and infinite conserved laws are presented by using TU-pattern. The algorithm involves wide applications for nonlinear evolution equations.

Formal Inferring the Law of Conservation of Energy from Assuming A-Priori-ness of Knowledge in a Formal Axiomatic Epistemology System Sigma

The research purpose is invention (construction) of a formal logical inference of the Law of Conservation of Energy within a logically formalized axiomatic epistemology-and-axiology theory Sigma from a precisely defined assumption of a-priori-ness of knowledge. For realizing this aim, the following work has been done: 1) a two-valued algebraic system of formal axiology has been defined precisely and applied to proper-philosophy of physics, namely, to an almost unknown (not-recognized) formal-axiological aspect of the physical law of conservation of energy;2) the formal axiomatic epistemology-and-axiology theory Sigma has been defined precisely and applied to proper-physics for realizing the above-indicated purpose. Thus, a discrete mathematical model of relationship between philosophy of physics and universal epistemology united with formal axiology has been constructed. Results: 1) By accurate computing relevant compositions of evaluation-functions within the discrete mathematical model, it is demonstrated that a formal-axiological analog of the great conservation law of proper physics is a formal-axiological law of two-valued algebra of metaphysics. (A precise algorithmic definition of the unhabitual (not-well-known) notion “formal-axiological law of algebra of metaphysics” is given.) 2) The hitherto never published significantly new nontrivial scientific result of investigation presented in this article is a formal logical inference of the law of conservation of energy within the formal axiomatic theory Sigma from conjunction of the formal-axiological analog of the law of conservation of energy and the assumption of a-priori-ness of knowledge.

Draw of Infinite Energy from Space and Negations of Two Important Laws

This paper shows that energy of 105 ton of oil can be obtained from space by fs (fermtosecond) eletromagnetic pulse technique in one second and one cm3 without any loss. This paper shows that the energy conservation law and Fermi golden rule should be negatived in some cases. The negation of Fermi golden rule has important influences on many fields based on quantum mechanics. For example, the present knowledge on the charge distribution in atomic nucleus might be wrong completely. This paper emphasizes that the proposition on introducing the concept of the energy support ability in space will cause a series of unimaginable discoveries, and, therefore is of epoch-making significance. This paper gives indirect experimental verifications for the necessity of introducing the concept of energy support ability in space, and suggests a very simple experiment to show directly that the energy conservation law and Fermi golden rule should be negatived in some cases.

Discovering exact,gauge-invariant,local energy–momentum conservation laws for the electromagnetic gyrokinetic system by high-order field theory on heterogeneous manifolds

Gyrokinetic theory is arguably the most important tool for numerical studies of transport physics in magnetized plasmas.However,exact local energy–momentum conservation laws for the electromagnetic gyrokinetic system have not been found despite continuous effort.Without such local conservation laws,energy and momentum can be instantaneously transported across spacetime,which is unphysical and casts doubt on the validity of numerical simulations based on the gyrokinetic theory.The standard Noether procedure for deriving conservation laws from corresponding symmetries does not apply to gyrokinetic systems because the gyrocenters and electromagnetic field reside on different manifolds.To overcome this difficulty,we develop a high-order field theory on heterogeneous manifolds for classical particle-field systems and apply it to derive exact,local conservation laws,in particular the energy–momentum conservation laws,for the electromagnetic gyrokinetic system.A weak Euler–Lagrange(EL)equation is established to replace the standard EL equation for the particles.It is discovered that an induced weak EL current enters the local conservation laws,and it is the new physics captured by the high-order field theory on heterogeneous manifolds.A recently developed gauge-symmetrization method for high-order electromagnetic field theories using the electromagnetic displacement-potential tensor is applied to render the derived energy–momentum conservation laws electromagnetic gauge-invariant.

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES(Ⅸ)—THERMOMECHANICS

The existing fundamental laws of thermodynamics for micropolar continuum field theories are restudied and their incompleteness is pointed out. New first and second fundamental laws for thermostatics and thermodynamics for micropolar continua are postulated. From them all equilibrium equations and the entropy inequality of thermostatics as well as all balance equations and the entropy rate inequalities are naturally and simultaneously deduced. The comparisons between the new results presented here and the corresponding results demonstrated in existing monographs and textbooks concerning micropolar continuum mechanics are made at any time. It should be emphasized to note that, the problem of why the local balance equation of energy and the local entropy inequality could not be obtained from the existing fundamental laws of thermodynamics for micropolar continua, is believed to be clarified.

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (X) -MASTER BALANCE LAW

Through a comparison between the expressions of master balance laws and the conservation laws derived by Noether＇s theorem, a unified master balance law and six physically possible balance equations for micropolar continuum mechanics are naturally deduced. Among them, by extending the well-known conventional concept of energymomentum tensor, the rather general conservation laws and balance equations named after energy-momentum, energy-angular momentum and energy-energy are obtained. It is clear that the forms of the physical field quantities in the master balance law for the last three cases could not be assumed directly by perceiving through the intuition. Finally, some existing results are reduced immediately as special cases.

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅳ)--SURFACE COUSERVATION LAWS

The purpose is to reestablish rather complete surface conservation laws for micropolar thermomechanical continua from the translation and the rotation invariances of the general balance law. The generalized energy-momentum and energy-moment of momentum tensors are presented. The concrete forms of surface conservation laws for micropolar thermomechanical continua are derived . The existing related results are naturally derived as special cases from the results proposed in this paper . The incomplete degrees of the existing surface conservation laws are clearly seen from the process of the deduction. The surface conservation laws for nonlocal micropolar thermomechanical continua may be easily obtained via localization .

The Basic Concepts and Basic Laws Relating to Matter and Gravitational Fields in Physics

In this work, the author applied the universal gauge field theory and Noether theorem to prove that universality exists for the Lorentz and Levi-Civita law of conservation of energy momentum tensor density. We also found that this conservation law has profound implications in physics. For example, based on this law, one can explore the origin of the matter field, and propose a new view about what is “dark energy” and what is “dark matter”.

Local structure-preserving methods for the generalized Rosenau-RLW-KdV equation with power law nonlinearity

Local structure-preserving algorithms including multi-symplectic, local energy- and momentum-preserving schemes are proposed for the generalized Rosenau-RLW-KdV equation based on the multi-symplectic Hamiltonian formula of the equation. Each of the present algorithms holds a discrete conservation law in any time-space region. For the original problem subjected to appropriate boundary conditions, these algorithms will be globally conservative. Discrete fast Fourier transform makes a significant improvement to the computational efficiency of schemes. Numerical results show that the proposed algorithms have satisfactory performance in providing an accurate solution and preserving the discrete invariants.

Riccati-type Backlund transformations of nonisospectral and generalized variable-coefficient KdV equations

We extend the method of constructing Bgcklund transformations for integrable equations through Riccati equations to the nonisospectral and the variable-coefficient equations. By taking nonisospectral and generalized variable-coefficient Korteweg-de Vries （KdV） equations as examples, their Backlund transformations are obtained under a more generalized constrain condition. In addition, the Lax pairs and infinite numbers of conservation laws of these equations are given. Es- pecially, some classical equations such as the cylindrical KdV equation are just the special cases of the constrain condition.

Use of Augmented Reality Methods to Support Legal Conflicts in the Planning Process for Wind Turbines Using the Example of the Landscape Conservation Area “Eulenkopf and Surroundings”

The world’s growing energy demand poses a serious problem. At the same time fossil fuels are finite, which we must work against. Therefore, the Federal Government of Germany has set itself the goal to push forward the use of renewable energy in order to completely do without the generation of nuclear energy by 2023. There are, however, no specific guidelines from the European Directive on the promotion of electricity from renewable energy sources for the internal electricity market regarding how high each share of the different production method should be and, above all, which specific aim should be achieved by the share of wind energy. Nevertheless, it presents a crucial step toward a nuclear phaseout and a concomitant change of course of the Federal Government of Germany in the spring of 2011 regarding the expansion of renewable energy, taking the nuclear catastrophe in Fukushima into account. Using new legal planning approaches, also including the area of Rhineland-Palatinate, opportunities should be provided to make previously protected land available for setting up facilities for the generation of renewable energy. However, it is important to examine the legal situation regarding the installation of these kinds of constructions more detailed, as no general statements can be made. This will be illustrated using the example of the landscape conservation area “Eulenkopf and surrounding area” in the district of Kaiserslautern. The stated goal of the Social Democrat/Green coalition of the federal state government of Rhineland-Palatinate is to considerably expand the generation of electricity from renewable energy sources so that by 2030 at least the entire electricity demand can be covered by those. Due to the enormous potential of wind power, it is therefore necessary to quintuple its share of electricity generation by 2020, compared to 2011 numbers. In order to achieve the desired political objectives, by 2030 the number of turbines has to be increased to around 2650, representing a capacity of 7500 MW. This increase gives reason for boundary conditions to manage the generation of wind energy to be adjusted. This is intended to facilitate management and simultaneously minimise negative effects, such as the “sprawling” of wind turbines.

The Laws of the Chemical Psychology

As a result of the research, new knowledge about the chemical mechanisms of mental processes in the brain was obtained,which allowed us to identify new patterns in chemical psychology. We proposed the laws of conservation of the quantity and energy of emotions.